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Mathematics 2016, 4(2), 39; doi:10.3390/math4020039

Morphisms and Order Ideals of Toric Posets

Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975, USA
Academic Editor: Michael Falk
Received: 1 January 2015 / Revised: 19 May 2016 / Accepted: 25 May 2016 / Published: 4 June 2016
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Abstract

Toric posets are in some sense a natural “cyclic” version of finite posets in that they capture the fundamental features of a partial order but without the notion of minimal or maximal elements. They can be thought of combinatorially as equivalence classes of acyclic orientations under the equivalence relation generated by converting sources into sinks, or geometrically as chambers of toric graphic hyperplane arrangements. In this paper, we define toric intervals and toric order-preserving maps, which lead to toric analogues of poset morphisms and order ideals. We develop this theory, discuss some fundamental differences between the toric and ordinary cases, and outline some areas for future research. Additionally, we provide a connection to cyclic reducibility and conjugacy in Coxeter groups. View Full-Text
Keywords: Coxeter group; cyclic order; cyclic reducibility; morphism; partial order; preposet; order ideal; order-preserving map; toric hyperplane arrangement; toric poset; 06A06, 52C35 Coxeter group; cyclic order; cyclic reducibility; morphism; partial order; preposet; order ideal; order-preserving map; toric hyperplane arrangement; toric poset; 06A06, 52C35
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Macauley, M. Morphisms and Order Ideals of Toric Posets. Mathematics 2016, 4, 39.

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